Analytic design of optimum holographic optical elements

A method is developed for analytically determining the holographic optical element phase function that is optimum for transforming a set of input wave fronts into a corresponding set of output wave fronts. These sets are allowed to be infinite in the sense that the wave-front phases can be given as functions of continuous parameters. The method can be tolerant of specified wave-front aberrations, with the optimum amount of these aberrations determined as part of the solution process. For many practical design problems, the phase function and its first derivatives will be continuous. The method is applied to the design of a one-dimensional Fourier-transform holographic element with the input wave-front angle of arrival as a continuous parameter and with the optimum distortion of the output plane determined by the solution. The resulting design compares favorably with other work using damped-least-squares optimization.